##### Inverse Functions

The inverse of any function $\displaystyle (f)$ is a function that will do the opposite of $\displaystyle f$. In other words the function that will undo the effects of $\displaystyle f$. So, if $\displaystyle f$ maps 4 into 13, then the inverse of $\displaystyle f$ will map 13 onto 4.

In effect, when $\displaystyle f$ is applied to a number and the reverse of $\displaystyle f$ is applied to the result, you will get back to the number you started with.

In simple cases, you can find the inverse of a function by inspection. For example, the inverse of $\displaystyle x\to x+5$ must be $\displaystyle x\to x-5$ because subtraction is the inverse of addition, to undo add five you have to subtract five.

Similarly, the inverse of $\displaystyle x\to 2x$ is $\displaystyle x\to \frac{x}{2}$, because to undo multiply by two you have to divide by two.

The inverse of the function $\displaystyle (f)$ is written as $\displaystyle {{f}^{{-1}}}$

So if$\displaystyle (f)=x+5$, then $\displaystyle {{f}^{{-1}}}(x)=x-5$ and if $\displaystyle g(x)=2x$ then$\displaystyle {{g}^{{-1}}}(x)=\frac{x}{2}$

Some functions do not have an inverse. Think about the function $\displaystyle x\to 2x$. This is a function because for every value of$\displaystyle x$, there is only one value of $\displaystyle {{x}^{2}}$. The inverse (in other words, the square root) is not a function because a positive number has two square roots, one negative, and one positive.

### Finding the inverse of a function

There are two methods of finding the inverse:

Method 1: Using a flow diagram

In this method you draw a flow diagram for the function and then work out the inverse by “reversing” the flow to undo the operations in the boxes.

Method 2: Reversing the mapping

In this method you use the fact that if $\displaystyle f$ maps $\displaystyle x$ onto $\displaystyle y$, then $\displaystyle {{f}^{{-1}}}$maps$\displaystyle y$  onto $\displaystyle x$. To find $\displaystyle {{f}^{{-1}}}$you have to find a value of  $\displaystyle x$ that corresponds to a given value of$\displaystyle y$.

Let’s see some examples using both methods.

Example 1: Find the inverse of$\displaystyle f(x)=3x-4$.

Using method 1, “the flow diagram” we get:

$\displaystyle f:input\to \times 3\to -4\to output$

$\displaystyle {{f}^{{-1}}}:output\leftarrow \div 3\leftarrow +4\leftarrow input$

Let $\displaystyle x$ be the input to $\displaystyle {{f}^{{-1}}}$

$\displaystyle \frac{{x+4}}{3}\leftarrow \div 3\leftarrow +4\leftarrow x$

$\displaystyle {{f}^{{-1}}}(x)=\frac{{x+4}}{3}$

Using method 2, “reversing the mapping”

Suppose the function maps $\displaystyle x$ onto $\displaystyle y$ ( $\displaystyle y$ is the subject). Make $\displaystyle x$ the subject of the formula, so that $\displaystyle y$ maps into $\displaystyle x$.

$\displaystyle y=3x-4$

$\displaystyle y+4=3x$

$\displaystyle x=\frac{{y+4}}{3}$

You  know that $\displaystyle {{f}^{{-1}}}$ maps $\displaystyle y$ onto $\displaystyle x$, so $\displaystyle {{f}^{{-1}}}(y)=\frac{{y+4}}{3}$

This is usually written in terms of $\displaystyle x$ so $\displaystyle {{f}^{{-1}}}(x)=\frac{{x+4}}{3}$

Example 2: Given $\displaystyle g(x)=5-2x$, find $\displaystyle {{g}^{{-1}}}(x)$.

Using method 1, “the flow diagram” you get:

$\displaystyle g:input\to \times (-2)\to +5\to output$

$\displaystyle {{g}^{{-1}}}:output\leftarrow \div (-2)\leftarrow -5\leftarrow output$

Let $\displaystyle x$ be the input to $\displaystyle {{g}^{{-1}}}$

$\displaystyle \frac{{x-5}}{{-2}}\leftarrow \div (-2)\leftarrow -5\leftarrow x$

$\displaystyle {{g}^{{-1}}}(x)=\frac{{x-5}}{{-2}}=\frac{{5-x}}{2}$

Using method 2, “reversing the mapping”

This means $\displaystyle g$ maps $\displaystyle x$ onto $\displaystyle y$

Make $\displaystyle x$ the subject of the formula so, that $\displaystyle y$ maps onto $\displaystyle x$.

Let $\displaystyle y=5-2x$

$\displaystyle 2x=5-y$

$\displaystyle x=\frac{{5-y}}{2}$

$\displaystyle {{g}^{{-1}}}$  maps  $\displaystyle y$  onto$\displaystyle x$, so  $\displaystyle {{g}^{{-1}}}(y)=\frac{{5-y}}{2}$

This is usually written in terms of $\displaystyle x$ so, $\displaystyle {{g}^{{-1}}}(x)=\frac{{5-x}}{2}$

Example 3: Given the function $\displaystyle g(x)=\frac{x}{3}-44$ find $\displaystyle {{g}^{{-1}}}(x)$.

Using method 1, “the flow diagram” you get:

$\displaystyle g:input\to \div 3\to -44\to output$

$\displaystyle {{g}^{{-1}}}:output\leftarrow \times 3\leftarrow +44\leftarrow output$

Let $\displaystyle x$ be the input to $\displaystyle {{g}^{{-1}}}$

$\displaystyle 3(x+44)\leftarrow \times 3\leftarrow +44\leftarrow x$

$\displaystyle {{g}^{{-1}}}(x)=3(x+44)=3x+132$

Using method 2, “reversing the mapping

This means $\displaystyle g$ maps $\displaystyle x$ onto $\displaystyle y$

Make $\displaystyle x$ the subject of the formula so, that $\displaystyle y$ maps onto $\displaystyle x$.

Let $\displaystyle y=\frac{x}{3}-44$

$\displaystyle y+44=\frac{x}{3}$

$\displaystyle x=3(y+44)$

$\displaystyle {{g}^{{-1}}}$  maps  $\displaystyle y$  onto$\displaystyle x$, so  $\displaystyle {{g}^{{-1}}}(y)=3(y+44)$

This is usually written in terms of $\displaystyle x$ so,$\displaystyle {{g}^{{-1}}}(x)=3(x+44)$